Download Arithmetic Series

U8L1: Arithmetic
Sequences and Series
EQ: What are arithmetic
sequences and series?
How do I find values
regarding them?
Arithmetic Sequences
Every day a radio station asks
a question for a prize of
$150. If the 5th caller
does not answer correctly,
the prize money increased
by $150 each day until
someone correctly answers
their question.
Arithmetic Sequences
Make a list of the prize
amounts for a week
(Mon - Fri) if the contest
starts on Monday and no one
answers correctly all week.
Arithmetic Sequences
• Monday :
• Tuesday:
• Wednesday:
• Thursday:
• Friday:
$150
$300
$450
$600
$750
Arithmetic Sequences
• These prize amounts form a
sequence, more specifically
each amount is a term in an
arithmetic sequence. To
find the next term we just
add $150.
Definitions
• Sequence: a list of numbers
in a specific order.
• Term: each number in a
sequence
Definitions
• Arithmetic Sequence: a
sequence in which each term
after the first term is
found by adding a constant,
called the common
difference (d), to the
previous term.
Explanations
• 150, 300, 450, 600, 750…
• The first term of our
sequence is 150, we denote
the first term as a1.
• What is a2?
• a2 : 300 (a2 represents the
2nd term in our sequence)
Explanations
• a3 = ?
a4 = ?
• a3 : 450 a4 : 600
a5 = ?
a5 : 750
• an represents a general term
(nth term) where n can be
any number.
Explanations
• Sequences can continue
forever. We can calculate as
many terms as we want as
long as we know the common
difference in the sequence.
Explanations
• Find the next three terms in
the sequence:
2, 5, 8, 11, 14, __, __, __
• 2, 5, 8, 11, 14, 17, 20, 23
• The common difference is?
• 3!!!
Explanations
• To find the common
difference (d), just subtract
any term from the term that
follows it.
• FYI: Common differences
can be negative.
Formula
• What if I wanted to find the
50th (a50) term of the
sequence 2, 5, 8, 11, 14, …?
Do I really want to add 3
continually until I get there?
• There is a formula for
finding the nth term.
Formula
• Let’s see if we can figure the
formula out on our own.
• a1 = 2, to get a2 I just add 3
once. To get a3 I add 3 to a1
twice. To get a4 I add 3 to
a1 three times.
Formula
• What is the relationship
between the term we are
finding and the number of
times I have to add d?
• The number of times I had
to add is one less then the
term I am looking for.
Formula
• So if I wanted to find a50
then how many times would I
have to add 3?
• 49
• If I wanted to find a193 how
many times would I add 3?
• 192
Formula
• So to find a50 I need to take
d, which is 3, and add it to
my a1, which is 2, 49 times.
That’s a lot of adding.
• But if we think back to
elementary school, repetitive
adding is just multiplication.
Formula
• 3 + 3 + 3 + 3 + 3 = 15
• We added five terms of
three, that is the same as
multiplying 5 and 3.
• So to add three forty-nine
times we just multiply 3 and
49.
Formula
• So back to our formula, to
find a50 we start with 2 (a1)
and add 3•49. (3 is d and 49
is one less than the term we
are looking for) So…
• a50 = 2 + 3(49) = 149
Formula
• a50 = 2 + 3(49) using this
formula we can create a
general formula.
• a50 will become an so we can
use it for any term.
• 2 is our a1 and 3 is our d.
Formula
• a50 = 2 + 3(49)
• 49 is one less than the term
we are looking for. So if I
am using n as the term I am
looking for, I multiply d by
n - 1.
Formula
• Thus my formula for finding
any term in an arithmetic
sequence is an = a1 + d(n-1).
• All you need to know to find
any term is the first term in
the sequence (a1) and the
common difference.
Example
• Let’s go back to our first
example about the radio
contest. Suppose no one
correctly answered the
question for 15 days. What
would the prize be on day
16?
Example
• an = a1 + d(n-1)
• We want to find a16. What is
a1? What is d? What is n-1?
• a1 = 150, d = 150,
n -1 = 16 - 1 = 15
• So a16 = 150 + 150(15) =
• $2400
Example
• 17, 10, 3, -4, -11, -18, …
• What is the common
difference?
• Subtract any term from the
term after it.
• -4 - 3 = -7
•d = - 7
Definition
• 17, 10, 3, -4, -11, -18, …
• Arithmetic Means: the
terms between any two
nonconsecutive terms of an
arithmetic sequence.
Arithmetic Means
• 17, 10, 3, -4, -11, -18, …
• Between 10 and -18 there
are three arithmetic means
3, -4, -11.
• Find three arithmetic means
between 8 and 14.
Arithmetic Means
• So our sequence must look
like 8, __, __, __, 14.
• In order to find the means
we need to know the common
difference. We can use our
formula to find it.
Arithmetic Means
• 8, __, __, __, 14
• a1 = 8, a5 = 14, & n = 5
• 14 = 8 + d(5 - 1)
• 14 = 8 + d(4)
subtract 8
• 6 = 4d
divide by 4
• 1.5 = d
Arithmetic Means
• 8, __, __, __, 14 so to find
our means we just add 1.5
starting with 8.
• 8, 9.5, 11, 12.5, 14
Additional Example
• 72 is the __ term of the
sequence -5, 2, 9, …
• We need to find ‘n’ which is
the term number.
• 72 is an, -5 is a1, and 7 is d.
Plug it in.
Additional Example
• 72 = -5 + 7(n - 1)
• 72 = -5 + 7n - 7
• 72 = -12 + 7n
• 84 = 7n
• n = 12
• 72 is the 12th term.
Section 11-2
Arithmetic
Series
Arithmetic Series
• The African-American
celebration of Kwanzaa
involves the lighting of
candles every night for
seven nights. The first night
one candle is lit and blown
out.
Arithmetic Series
• The second night a new
candle and the candle from
the first night are lit and
blown out. The third night a
new candle and the two
candles from the second
night are lit and blown out.
Arithmetic Series
• This process continues for
the seven nights.
• We want to know the total
number of lightings during
the seven nights of
celebration.
Arithmetic Series
• The first night one candle
was lit, the 2nd night two
candles were lit, the 3rd
night 3 candles were lit, etc.
• So to find the total number
of lightings we would add:
1+2+3+4+5+6+7
Arithmetic Series
• 1 + 2 + 3 + 4 + 5 + 6 + 7 = 28
• Series: the sum of the terms
in a sequence.
• Arithmetic Series: the sum
of the terms in an arithmetic
sequence.
Arithmetic Series
• Arithmetic sequence:
2, 4, 6, 8, 10
• Corresponding arith. series:
2 + 4 + 6 + 8 + 10
• Arith. Sequence: -8, -3, 2, 7
• Arith. Series: -8 + -3 + 2 + 7
Arithmetic Series
• Sn is the symbol used to
represent the first ‘n’ terms
of a series.
• Given the sequence 1, 11, 21,
31, 41, 51, 61, 71, … find S4
• We add the first four terms
1 + 11 + 21 + 31 = 64
Arithmetic Series
• Find S8 of the arithmetic
sequence 1, 2, 3, 4, 5, 6, 7, 8,
9, 10, …
•1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 =
• 36
Arithmetic Series
• What if we wanted to find
S100 for the sequence in the
last example. It would be a
pain to have to list all the
terms and try to add them
up.
• Let’s figure out a formula!! :)
Sum of Arithmetic Series
• Let’s find S7 of the sequence
1, 2, 3, 4, 5, 6, 7, 8, 9, …
• If we add S7 in too different
orders we get:
S7 = 1 + 2 + 3 + 4 + 5 + 6 + 7
S7 = 7 + 6 + 5 + 4 + 3 + 2 + 1
2S7 = 8 + 8 + 8 + 8 + 8 + 8 + 8
Sum of Arithmetic Series
S7 = 1 + 2 + 3 + 4 + 5 + 6 + 7
S7 = 7 + 6 + 5 + 4 + 3 + 2 + 1
2S7 = 8 + 8 + 8 + 8 + 8 + 8 + 8
2S7 = 7(8) 7 sums of 8
7
S7 = /2(8)
Sum7of Arithmetic Series
• S7 = /2(8)
• What do these numbers
mean?
• 7 is n, 8 is the sum of the
first and last term (a1 + an)
• So Sn = n/2(a1 + an)
Examples
• Sn = n/2(a1 + an)
• Find the sum of the first 10
terms of the arithmetic
series with a1 = 6 and a10 =51
• S10 = 10/2(6 + 51) = 5(57) =
285
Examples
• Find the sum of the first 50
terms of an arithmetic
series with a1 = 28 and d = -4
• We need to know n, a1, and
a50.
• n= 50, a1 = 28, a50 = ?? We
have to find it.
Examples
• a50 = 28 + -4(50 - 1) =
28 + -4(49) = 28 + -196 =
-168
• So n = 50, a1 = 28, & an =-168
• S50 = (50/2)(28 + -168) =
25(-140) = -3500
Examples
• To write out a series and
compute a sum can
sometimes be very tedious.
Mathematicians often use
the greek letter sigma &
summation notation to
simplify this task.
Examples
last value of n
5
𝑛+1
𝑛=1
formula used to
find sequence
First value of n
• This means to find the sum
of the sums n + 1 where we
plug in the values 1 - 5 for n
Examples
5
𝑛+1
𝑛=1
• Basically we want to find
(1 + 1) + (2 + 1) + (3 + 1) +
(4 + 1) + (5 + 1) =
•2 + 3 + 4 + 5 + 6 =
• 20
• So
Examples
5
𝑛 + 1 = 20
𝑛=𝟏
• Try:
7
3𝑥 − 2
𝑥=2
• First we need to plug in the
numbers 2 - 7 for x.
7
Examples
3𝑥 − 2
𝑥=2
• [3(2)-2]+[3(3)-2]+[3(4)-2]+
[3(5)-2]+[3(6)-2]+[3(7)-2] =
• (6-2)+(9-2)+(12-2)+(15-2)+
(18-2)+ (21-2) =
• 4 + 7 + 10 + 13 + 17 + 19 = 70